Euler defines *magnitude* or *quantity*.
He says that a magnitude is something that can increase or decrease.
Increase or decrease are two aspects of change. And these types of change occurs through time.
In essence, we have, from most to less fundamental:
time ---> increase/decrease ---> magnitude
What happens when we want to consider time as a magnitude? Then, the previous chain becomes a loop
When dealing with very fundamental concepts, they become fuzzy and they easily overlap. For example, try to define the concept of life and see where this gets you.
But this is not the place to discuss such a thing. Let's continue.
Examples of magnitudes/quantities: a length, an area, a weight... These are physical magnitudes. We can also have chapters of a book, unfinished projects, types of curves, etc.
There are many different types of magnitudes. As many as you can imagine. Euler says that each kind of magnitude originates a different branch of mathematics. I am not sure about this.
He also says that mathematics is the science of quantity, or the science investigating the means of measuring quantity.
This is interesting, since I have witnessed many times a debate of whether mathematics is a science or not. It seems clear that we define science as discipline in which you have theories and models but where measuring is what decides which models or theories are refuted. In physics it is clear that we measure, but what about mathematics? Do we measure?
For example, we could "measure" the zeros in Riemann's zeta function. Is this a true measurement? I don't have an answer for this.
Perhaps Euler, by "science of quantity" means that mathematics does not measure like a natural science but that it is concerned instead in the concept of quantity itself. An analogy would be some craft in which you use tools, and then you need tools to craft these tools, or at least to understand them.
Beware of the word "quantity". I would prefer to use only "magnitude", since now we are going to distinguish between "magnitude" and "number" and the term quantity will be used to address both of them. Up to now, Euler has used the term quantity as a synonym of magnitude.
We cannot measure a magnitude in absolute terms. This is a meaningless thing to even think. What is meaningful is the mutual relation between two magnitudes.
For example, we have two lines and we want to measure their length. So we can say that line A is 2 times line B or that line B is 0.5 times line A. Notice that in order to do so, the types of magnitude must be the same. Otherwise we cannot do such mutual relation.
Imagine now, instead of having two lines, we have many of them. Imagine the sheer amount of mutual relations we could define between them! No need to complicate it so much. Just choose one line, for example, line u, and arbitrarily call the "unit". So that the rest of lines can have a mutual relation with respect to line u. For example, line A could be 3 times line u, line B could be 4 times line u, etc.
line u --- line A --------- line B ------------
Having a universal magnitude as the reference or unit is a very practical thing. Think, for example, of the meter for length, or the second for time. They are completely arbitrary units, but very useful. Notice as well how annoying is to go to different countries in which they use different mass or length units. An interesting thing here is to observe that time units are universal through the entire planet, as far as I know. Why is time so well unified while length and mass are not?
Once we have a mutual relation between two magnitudes, say A and u, we can write
A = 2 u ,
or we can write
A / u = 2
The first way of writing, A = 2 u, is our typical way of expressing it, but the second way of writing allows us to explicitly appreciate the mutual relation between the two magnitudes A and u as a *ratio* or a *proportion*.
Once we get this, Euler proceeds to define *number* as the proportion between two magnitudes of the same type. So this second way of writing not only allows us to appreciate the mutual relation between magnitudes: it also allows to define the concept of number itself as exactly this ratio!
In essence, when we write expressions as we are used to, recall what is what:
A = 2 u magnitude number magnitude to be of measured reference
It appears that all magnitudes may be expressed by numbers, says Euler. Isn't this a magnificent thing, almost a miracle?
He also defines Algebra as the science of numbers. He also calls it Analysis, but the modern meaning of this word does not correspond to a synonym of Algebra, so let's stick to the term Algebra.
This means that from now on, in this text, we will forget about magnitudes and we will deal only with numbers. And, as I said before, we will use the term quantity to refer to numbers. How can we be so sloppy?
Well, not so. Imagine that you define your unit magnitude as u = 1. This means that we are defining a kind of mathematical unit that allows us to identify magnitude = quantity = number. Just notice that this is only true for this particular definition of magnitude of reference (u=1).